Open. The site's proof-claims tab carries two partial proof claims by Ricky Cipollini, both attributing the proof to the model GPT-5.6 Sol and both pointing to one write-up on a collaborative editing service: an upper bound, submitted 2026-07-27, of the shape $\log n_k\le (1+o(1))\,k\log\log k/\log k$ with explicit second-order terms, recorded on its claim page (Cipollini, 2026), and a lower bound, submitted 2026-08-04, of the shape $\log n_k\ge c(\log k)^2$, recorded on its claim page (Cipollini, 2026). Neither of Cipollini's claims settles the problem, and neither has acceptance evidence; the site's label is unchanged (OPEN; page last edited 01 February 2026), and the claims are recorded without being adopted (proof-claims thread accessed 2026-10-07). Two further upper bounds lie outside the tab. Patrick White's repository of 25 July 2026 claims $n_k\le\exp(Ck\log\log k/\log k)$ for large $k$; it is recorded on its claim page (White, 2026). A Lean 4 proof of 1 October 2026 by the LEAP prover agent, linked by the formal-conjectures catalog, gives the stronger bound $n_k=O(\exp(\frac{k}{\log k}(\log\log k+\log\log\log k+\log2)))$; it is recorded on its claim page (LEAP, 2026). Monier's published bound $n_k\le k!$ for $k\ge3$ (Amer. Math. Monthly 1985), which the site's commentary credits, is an accepted partial claim on its claim page (Monier, 1985).